Two and Three Dimensional Problems: Atoms and molecules have three dimensional properties. We will therefore need to treat the Schrödinger equation for three dimensional cases. Examples of three dimensional problems include the H atom Hamiltonian (electronic energies), and the rotation of a molecule in three dimensions.
PIAB Calculations: Last day we performed a simple, one dimensional PIAB calculation for trans- butadiene (H 2 C=CH-CH=CH 2 ) and estimated that the lowest frequency electronic absorption would be found at 45,400 cm -1. This agrees extremely well (fortuitously so) with the experimental value of 46,100 cm -1.
PIAB Calculations – Limitations:
Orthogonal Vectors and Wave Functions: In many mathematics courses the use of the Cartesian axes system has been discussed. Here, three axes are oriented at 90 0 angles in space. Such a choice greatly simplifies all sorts of mathematical calculations. Vectors: In three dimensions a vector ρ might be represented as ρ = (x 1, y 1, z 1 )
Orthonormal Vectors:
Orthonormal Vectors: On the previous slide the vector ρ 1 is called a unit vector. Vectors are termed orthogonal if their dot products are zero, i.e. if ρ 1 ∙ρ 2 = 0. Example of orthogonal vectors: ρ 1 = (2, 0, 0), ρ 2 = (0, 2, 3) A triad of three unit vectors that are othonormal enables all other vectors to be described as linear combinations of the three orthonormal unit vectors.
Orthonormal Vectors: For a simple Cartesian axis system the three required unit vectors are clearly: ρ 1 = (1, 0, 0), ρ 2 = (0, 1, 0) and ρ 3 = (0, 0, 1). Any other vector can now be expressed as a linear combination of the unit vectors. Eg. if ρ 4 = (3, -1.5, 4.4) then ρ 4 = 3ρ ρ ρ 3
Orthonormal Wave Functions:
Orthonormal Wave Functions:
PIAB Wave Functions are Orthonormal: The PIAB wave functions, at least when normalized, comprise an orthonormal set. The various wave functions (for the n=1,2,3…… cases in one dimension) have different eigenvalues. This is true for other cases/Hamiltonians. We will revisit this when we further consider the postulates of quantum mechanics.
PIAB – Three Dimensions:
PIAB – Three Dimensions:
PIAB – Three Dimensions:
PIAB – Three Dimensions: In the previous slide we have dispensed with the partial derivatives previously present in the Hamiltonian. This follows from the assumption that the wave function cane be “factored” or that variables can be “separated”. This is reasonable as long as the particle is found within the 3-dimensional box where V(x,y,z) = 0.
PIAB – Three Dimensions:
PIAB – Three Dimensions:
PIAB – 3 Dimensions – Energies:
PIAB Models – Key Features: 1. For cases of all dimensions energy level spacings get smaller as the box gets bigger and as the particle gets heavier. This trend will be seen again when we discuss rotational spectra and vibrational spectra. This trend is very important in any consideration of statistical mechanics (properties of individual molecules → properties of systems).
PIAB Models – Key Features: 2. An n value of zero is impossible. Why? Thus, the PIAB has kinetic energy even at absolute zero since it can “fall” no further than the n = 1 level. Something similar is seen for vibrational energies. Beach Boys? 3. In two and three dimensions the “box” can have symmetry. A 2 dimensional box can, for example, be a rectangle or a square. Symmetry “simplifies” energy calculations.
PIAB Models – Key Features: 4. With symmetry we encounter for the first time (since first year!) the concept of degeneracy. We can have two (or more) energy levels, specified with a different set of quantum numbers, having the same energy.
Class Examples: We will construct in class energy ladders for one and higher dimension PIABs. These calculations will include the PIAB in a square box and the PIAB in a cubic box – where degeneracy is very important.